Abstract

Abstract. Let A be a finite algebra with a majority term. We characterize those algebras categorically equivalent to A. The description is in terms of a derived structure with universe consisting of all subalgebras of A × A, and with operations of composition, converse and intersection. The main theorem is used to get a different sort of characterization of categorical equivalence for algebras generating an arithmetical variety. We also consider clones of co-height at most two. In addition, we provide new proofs of several characterizations in the literature, including quasi-primal, lattice-primal and congruence-primal algebras. Majority operations have long held a special place in universal algebra. It has been known for quite some time that any variety of algebras possessing a majority term is congruence distributive. In 1975, Baker and Pixley discovered that for a finite algebra A with a majority term, the set of subalgebras of A2 completely determines the term operations on A. In addition, every subalgebra of Ak (with k ≥ 2) is completely determined by all of its 2-fold projections. Conversely, G. Bergman

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